How Euclidean distance algorithm calculate two different face images are match or not match in face recognition?
Simply put, Euclidean distance measures how far away two items are (see Neil Slater's comment).

In order to apply this to a pattern recognition task, you will need to convert the items to compare (in your case images of faces) into feature vectors (ie lists of numerical values), and then you do a pairwise comparison to work out how distant two faces are. You will then need to set a threshold where you treat two images as being the same face (typically where the distance between the feature vectors is small).

Selecting the right features is obviously crucial here. I'm not an expert on face descriptions, but it would probably be something like


the distance between the eyes
the distance between the bottom of the nose and the top of the upper lip
...


Once you have these measurements, store the values for each image in a vector and you can apply the Euclidean distance. Effectively, each image is represented as a point in the  $m$-dimensional feature space, where each measurement is a dimension. To select good features make sure they are not correlated (ie they are independent of each other) and are the same scale (eg all are distances, so not eye colour)

The choice of Euclidean distance is fairly minor: there are other distance metrics which might work equally well or even better. As I mentioned, the key point is selecting appropriate feature values.